To demonstrate the construction of an order four multirate scheme, we use as underlining schemes REF  the classical Runge-Kutta (RK4) scheme [1]} which we refer to as
MR-RK4.
With multirate factor \(m=20\) , we solve the ODE system (REF ) for \(n=10\)  numerically on the time interval \([0,40]\)  with masses \(\left(m_1,m_2\right) = \left(1,20\right)\) , spring constants \(\left(k_1,k_2\right) = \left(20,1\right)\)  and initial values
\(\begin{pmatrix}x_1(0),\;\dot{x}_1(0)\end{pmatrix} = \begin{pmatrix}-0.005,\;0\end{pmatrix}, \quad \begin{pmatrix}x_i(0),\;\dot{x}_i(0)\end{pmatrix} = \begin{pmatrix}0.1,\;0\end{pmatrix}, \quad \forall i=2,\ldots ,n.\) 
